We consider the scattering problem for the nonlinear kleingordon equation. Kleingordon equation the motion of a stretched vortex filament combines both selfinduction and elasticity. Separation of variables and exact solution of the klein. Lorentz invariance of kleingordon lagrangian physics forums. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. Physics, particle physics and fields, specific theories and interaction models, fundamental equations, kleingordon equation, physics publisher on behalf of the author collection flooved. We follow and modify the feshbachvillars formalism by separating the kleingordon equation into two coupled timedependent. So, none of the objections that led to discarding the klein equation held up with time.
Pdf numerical solution of kleingordon and sinegordon. Numerical solution of kleingordon and sinegordon equations using the numerical method of gridless lines. We solve the relativistic klein gordon equation for a light particle gravitationally bound to a heavy central mass, with the gravitational interaction prescribed by the metric of a spherically symmetric spacetime. The kleingordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field. We solve the goursat problem for fixed angular momentum n of the field with the restriction that n is non zero in the case of a massless field. Parallel spectral numerical methodsthe kleingordon. Remark on the scattering problem for the klein gordon equation with power nonlinearity.
This equation, studied in greater detail by klein and gordon whose name it now bears. Well, the problem with this idea in the case of klein gordan equation is that you end up with something that is not a continuity equation, so, if you try to use it as a probability density, you end up with something that have nonconstant normalization, and, thus, cant be used to define a probability distribution. Kleingordon equation a relativistic that is satisfying the requirements of the theory of relativity quantum. Kleingordon equation the behavior of the kleingordon equation. We follow and modify the feshbachvillars formalism by separating the kleingordon equation into two coupled timedependent schrodinger equations for particle and.
Stationary solutions of the kleingordon equation in a potential field. It is a quantized version of the relativistic energymomentum relation. Pdf kleingordon equation in hydrodynamical form researchgate. The plane waves 10 are basic solutions and the field 9 is constructed by. The derivation of the kleingordon equation 3, 4, 5, 6, 7is essentially identical to that of the schrodinger equation namely, an action ansatz is substituted in. Such solutions if realvalued correspond to certain constant mean curvature surfaces in euclidean 3space. Therefore, a system that can be described by a complex solution to the klein gordon equation also be described by a system of two independent particles with equal mass that have real solutions to the klein klein gordon equation. China weimin zhouy and jun yanz department of physics, new york university, 4 washington place, new york, ny, 3 a particle is always not pure. We solve the relativistic kleingordon equation for a light particle gravitationally bound to a heavy central mass, with the gravitational interaction prescribed by the metric of a spherically symmetric spacetime.
We explore kleingordon equation in the new framework of quasihermitian quantum mechanics. Integral transform approach to solving kleingordon. Mod01 lec01 introduction, the kleingordon equation youtube. A nekhoroshev type theorem for the nonlinear kleingordon. Its solutions include a quantum scalar or pseudoscalar field, a field whose. Kleingordon equation article about kleingordon equation. Chapter propagators of the real klein gordon field.
The supersymmetric approach is employed to calculate any. The klein gordon equation can be expressed in general possible form by. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with a linear transition from constant to zero density. Later, the equation was revived and is now known as kleingordon equation. This book develops a spectral theory for the integrable system of 2dimensional, simply periodic, complexvalued solutions u of the sinhgordon equation. Equation 12 is of course the relativistic energymomentum relation all. For a free particle, the equation is p p m 0c2 0 1 where p is the fourmomentum operator. Consider a nonlinear kleingordon equation on the unit circle, with smooth data of size. Pdf the feynman problem for the kleingordon equation. Bound state solutions of the kleingordon equation for strong potentials. The nonlinear term of the equation behaves like a power term. The kleingordon equation with a scalar potential and a vector potential in ddimensions reads where represents the spatial dimensionality and represents the laplace operator in ddimensions, is the reduced planck constant, and are the speed of light and relativistic energy of the system, respectively. As we can see now, the apparent problems were manifestations of the new physics. Sep 14, 2015 solving the klein gordon equation is quite similar to solving schrodingers equation.
Fok as a wave equation under the conditions of cyclicity in the fifth coordinate and was shortly afterwards deduced by several authors for example, w. The goursat problem at the horizons for the kleingordon. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with. A spectral theory for simply periodic solutions of the. Kleingordondirac equation, plane harmonic waves with di.
Klein gordon eld is characterized by \particles with rest mass mand no other structure e. Fourier transforming the klein gordon equation physics. Introduction to dispersive waves, dispersion relations, uniform kleingordon equation, linear superposition and the fourier transform, the stationaryphase method for linear dispersive waves 20 extra lecture pdf. Gordon equation of the symmetric generalized woods. Comparison among kleingordon equation, dirac equation. We consider a modified kleingordon equation that arises at ultra high energies.
Comparison among kleingordon equation, dirac equation and. Under minimal substitution the wave equation becomes quadratic in both the momentum and the coordinates,viz. Logical inference approach to relativistic quantum mechanics. At the present time, there have been no numerical studies of blow up of solutions to this equation without the assumption of radial symmetry. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. By introducing the concept of system mass into the klein gordon equation with the scalar and vector potentials, we prove that if the. We consider the kleingordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry, which is. Mod01 lec01 introduction, the kleingordon equation. If you like you can view the kleingordon equation as a \toy model for the maxwell equations. For an isolated filament the solution must have the form of a helix 7, or 3, although a correction to the frequency must be made. Notice that the lefthand part of the dynamic equation 4 has the. Consider a meson bound in a point nucleus with in nite mass mn. An overview over the kg propagators on minkowski spacetime is given in mikica kocic, invariant commutation and propagation functions invariant commutation and propagation functions, 2016 pdf. The exact energy eigenvalues and wavefunctions are derived analytically by using the nikiforov and uvarov method.
Klein gordon equation article about klein gordon equation. Cryptohermitian approach to the kleingordon equation. Soliton solutions of the generalized kleingordon equation. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. B c2 would decrease without lower bound if the charge number of nucleus zis su ciently large. The latter equation represent the conservation of the number of. Jun 19, 2014 mod01 lec01 introduction, the kleingordon equation nptelhrd. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. The f eynman problem for the klein gordon equation. Numerical analysis for the kleingordon equation with mass. The solution just like in the case with the wave function in the schrodinger equation just comes out to be some scalar function of x, y, z, and t. Kleingordon equation for quark pairs in color superconductor by.
The general solutions found, could be used for future explorations on the study for other speci. May 14, 2011 we derive the vortex dynamics for the nonlinear maxwellkleingordon equation with the ginzburglandau type potential. Therefore, the kleingordon equation was at first rejected. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. Dirac equation gordon equation dependent mass pseudospin symmetry spinless particle. Dispersive wave solutions of the kleingordon equation in. Bound state solutions for the swave are found by separating the kleingordon equation in two parts, unlike the similar works in the literature. Oskar kleins tomb at judiska norra begravningsplatsen in solna grey stone to the right. For comparison, we have also discussed the dirac equation. Bound state solutions of the kleingordon equation for strong. In this article, we derive the scalar klein gordon equation from the. If the inline pdf is not rendering correctly, you can download the.
Numerical solution of kleingordon and sine gordon equations using the numerical method of gridless lines. Here, kleingordon equation connecting repositories. A nonlinear kleingordon equation on star graphs nataliia goloshchapov a department of mathematics, imeusp, r ua do mat. The kleingordon equation can be expressed in general possible form by. Jan 30, 2008 we consider a new type of linear interaction in the klein gordon equation. New transform iterative method for solving some klein. Therefore, as long as energy and momentum follows the einsteins relation eq. Klein gordon equation f or a free relativistic particle the total energy e is no longer giv en b y the equation w e used to deriv e the sc hro edinger equation. For that reason it is also called the kleingordon field. These equations appear in the study of relativistic and quantum physics. So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. The dirac equation can be thought of in terms of a square root of the kleingordon equation. Gordon scalar field in the exterior of the domain of influence of a compact set.
So the kleingordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. Physics letters a vol 384, issue 12, 30 april 2020. That is where cryptohermitian approach can be applied. Vortex dynamics for the nonlinear maxwellkleingordon equation. Analytical solutions of the kleingordon equation with a. The relativisticallyinvariant quantum equation describing spinless scalar or pseudoscalar particles, for example, and mesons.
The klein gordon equation 21 is completely separable in 3, therefore we look for its solution in the form. The kleingordon equation is named after oskar klein and walter gordon. We seek to introduce a mathematical method to derive the kleingordon equation and a set of relevant laws strictly. Solving the fractional nonlinear kleingordon equation by.
Pdf a nonlinear kleingordon equation on star graphs. On the basis of this description, we can describe in more detail the motion of such a particle, either by the kleingordon equation, or as a component of the spin value of the particle. Solutions to common problems with probability interpretation and inde. Stationary solutions of the kleingordon equation in a. Comparison among kleingordon equation, dirac equation and relativistic stationary schr. The eigenfunctions of the system are given and the degeneracy of the energy eigenvalues is discussed. When schrodinger developed his nonrelativistic wave equation, he also proposed a relativistic generalization. Pdf notes on the kleingordon equation researchgate. Delort universite paris, institut galilee cnrs, umr 7539, laboratoire analyse geom etrie et applications 99, avenue j. However, the situation becomes quite di erent in the theory of special relativity. Bound state solutions of kleingordon equation with the kratzer. Kleingordon eld is characterized by \particles with rest mass mand no other structure e. Introduction the problem has originally arisen from the paper 1 and the more later one 2, where the kleingordondirac equation kgd, i. The klein gordon equation is a relativistic wave equation valid for spinless particles.
Greens functions in physics version 1 uw faculty web. Dec 24, 2004 quadratic nonlinear kleingordon equation in 2d, cauchy problem hayashi, nakao and naumkin, pavel i. Pdf in this article, we derive the scalar parametrized kleingordon equation from the formal information theory framework. Theoretical physics 3 phys3661 quantum mechanics epiphany 2015 alexander lenz ippp, durham university april 8, 2015 1. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. Weisskopf discovered a suitable interpretation for the equation within the scope of quantum field theory. Mechanical interpretation of the kleingordon equation. Solution to kleingordon equation physics stack exchange. Relativistic quantum mechanics kleingordon equation dirac. Historically, the kleingordon equation was the first relativistic equation in quantum mechanics for the wave function. The energy equation and the corresponding unnormalized wave function are obtained analytically. Solutions of the klein gordon equation with generalized. New transform iterative method for solving some kleingordon.
Numerical computation of kleingordon equations arising in. View kleingordon equation research papers on academia. For the kleingordon equation, however, there is an exact solution. It is secondorder in space and time and manifestly lorentzcovariant. On the basis of this description, we can describe in more detail the motion of such a particle, either by the klein gordon equation, or as a component of the spin value of the particle. Solitary waves for nonlinear kleingordonmaxwell and. The nonrelativistic equation for the energy of a free particle is.
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